Summary
The problem of solidification of two semi-infinite materials with arbitrarily prescribed initial conditions is studied. This is different from the classical Stefan problem; there are no prescribed boundary conditions. It is found that there are, depending on the prescribed initial conditions, four different possibilities: (i) solidification starts immediately and the interfacial boundary is of the order oft 1/2 ast→0, (ii) solidification also starts immediately but the interfacial boundary is of an order other thant 1/2, (iii) no solidification will ever occur, and (iv) there is a pre-solidification period during which the temperatures of both materials are redistributed and solidification will occur only after the surface temperature has reached the freezing point. Conditions for the occurrence of these cases are examined and established. The exact solutions for each of these cases are derived and discussed.
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Abbreviations
- a n b n c n d n :
-
series coefficients of the temperature solutions, Eq. (2.7)
- k :
-
coefficient of thermal conductivity
- s(t) :
-
interfacial position
- x, t :
-
space and time variables
- u n v r :
-
series coefficients of the initial data, Eq. (2.4)
- A m n B m n :
-
constants defined by Eq. (2.12)
- C :
-
specific heat
- E n(z)F n(z):
-
functions in the error integral family, Eq. (2.5)
- G n(z):
-
polynomial in the error integral family, Eq. (2.5)
- I n(z)J n(z):
-
functions in the error integral family, Eq. (3.5)
- K(x, t; α):
-
fundamental solution of the heat equation, Eq. (3.6)
- L :
-
latent heat
- T(x, t) :
-
temperature
- U(x), V(x) :
-
initial data
- α:
-
coefficient of thermal diffusibility
- β r :
-
multinomial coefficient
- λ n :
-
series coefficient of the interfacial position
- µ i :
-
(k i /k II)ω i ,i=I,III
- ϱ:
-
density
- η:
-
\( = s/\sqrt {4\alpha t} \)
- ξ:
-
\( = x/\sqrt {4\alpha t} \)
- τ:
-
t 1/2
- ω i :
-
\( = \sqrt {\alpha _{II} /\alpha _i ,} = I,III\)
- Δ, Ω:
-
constants defined by Eq. (3.1)
- Φ n (z):
-
iterated derivative of the error integral
- f :
-
freezing point
- q :
-
initial state
- p :
-
pre-solidification period
- s :
-
surface atx=0
- I:
-
solid phase
- II:
-
newly formed solid phase
- III:
-
liquid phase
- ∼:
-
pre-solidification state
- *:
-
solidification state after pre-solidification
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Tao, L.N. The Cauchy-Stefan problem. Acta Mechanica 45, 49–64 (1982). https://doi.org/10.1007/BF01295570
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DOI: https://doi.org/10.1007/BF01295570